A characterization of partition polynomials and good Bernoulli trial measures in many symbols

Consider an experiment with d+1 possible outcomes, d of which occur with probabilities $x₁,..., x_{d}$. If we consider a large number of independent occurrences of this experiment, the probability of any event in the resulting space is a polynomial in $x₁,..., x_{d}$. We characterize those polynomials which arise as the probability of such an event. We use this to characterize those x⃗ for which the measure resulting from an infinite sequence of such trials is good in the sense of Akin.